The above equation is very special in the sense that it can be derived from various distinct approaches. This equation can be achieved by considering a quantum corrected entropy-area relation of \( S = \frac{A}{4} - \alpha \ln \left( \frac{A}{4} \right) \)  where A is the area of the apparent horizon and \(\alpha \)  is a dimensionless positive constant determined by the conformal anomaly of the fields, where the conformal anomaly is interpreted as a correction to the entropy of the apparent horizon [1]} . Also, this could be derived in terms of spacetime thermodynamics together with a generalized uncertainly principle of quantum gravity[2]}. This Friedmann equation can also be derived by considering an anti-de Sitter-Schwarzschild black hole via holographic renormalization with appropriate boundary conditions [3]}. Finally, a Chern-Simons type of theory can also yield this Friedmann equation [4]}. Hence the equation (7) can derived by a wide range of approaches to gravitational physics and can be representative of the effects of these different theories on the cosmological dynamics. In this way, besides the RS-II Braneworld and F(R) gravity theory discussed before, we are taking into account effects on cosmology from modified area-entropy relations, generalized uncertainty principles and holographic renormalization as well besides Chern-Simons gravity as well.
